Effects of tangential-type boundary condition discontinuities on the accuracy of the lattice Boltzmann method for heat and mass transfer.

We present a systematic study on the effects of tangential-type boundary condition discontinuities on the accuracy of the lattice Boltzmann equation (LBE) method for Dirichlet and Neumann problems in heat and mass transfer modeling. The second-order accurate boundary condition treatments for continuous Dirichlet and Neumann problems are directly implemented for the corresponding discontinuous boundary conditions. Results from three numerical tests, including both straight and curved boundaries, are presented to show the accuracy and order of convergence of the LBE computations. Detailed error assessments are conducted for the interior temperature or concentration (denoted as a scalar ϕ) and the interior derivatives of ϕ for both types of boundary conditions, for the boundary flux in the Dirichlet problem and for the boundary ϕ values in the Neumann problem. When the discontinuity point on the straight boundary is placed at the center of the unit lattice in the Dirichlet problem, it yields only first-order accuracy for the interior distribution of ϕ, first-order accuracy for the boundary flux, and zeroth-order accuracy for the interior derivatives compared with the second-order accuracy of all quantities of interest for continuous boundary conditions. On the lattice scale, the LBE solution for the interior derivatives near the singularity is largely independent of the resolution and correspondingly the local distribution of the absolute errors is almost invariant with the changing resolution. For Neumann problems, when the discontinuity is placed at the lattice center, second-order accuracy is preserved for the interior distribution of ϕ; and a "superlinear" convergence order of 1.5 for the boundary ϕ values and first-order accuracy for the interior derivatives are obtained. For straight boundaries with the discontinuity point arbitrarily placed within the lattice and curved boundaries, the boundary flux becomes zeroth-order accurate for Dirichlet problems; and all three quantities, including the interior and boundary ϕ values and the interior derivatives, are only first-order accurate for Neumann problems.